Difficult problem

dch

May 2010
4
0
Suppose n children holding loaded water pistols are standing in an open field, no three of them in line such that all the distances between pairs of them are distinct. At a given signal, each child shoots the child closest to him or her with water. show that if n is an even number then it is possible (but not necessarily the case) that every child gets wet. Show that if n is odd, then necessarily at least one child stays dry,

Thank you
 

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MHF Hall of Honor
Mar 2010
2,340
821
Chicago
Suppose n children holding loaded water pistols are standing in an open field, no three of them in line such that all the distances between pairs of them are distinct. At a given signal, each child shoots the child closest to him or her with water. show that if n is an even number then it is possible (but not necessarily the case) that every child gets wet. Show that if n is odd, then necessarily at least one child stays dry,

Thank you
The even part is really easy. Just imagine a scenario where you have pairs of children such that the two children in a pair are very close to each other, but each pair is far away from any other pair.

If you need to prove the part in parentheses, "but not necessarily the case," then look for a counterexample with small n. I found one for n = 4 in a few seconds..
 
Sep 2009
32
4
Milwaukee, Wisconsin
Maybe I am misunderstanding the question, but what if three kids stand in an isosceles triangle? All three kids could get wet in theory... yes?
 
Oct 2009
4,261
1,836
Maybe I am misunderstanding the question, but what if three kids stand in an isosceles triangle? All three kids could get wet in theory... yes?

I think you misunderstood: in this case there are two pairs of children at the same distance...

Tonio
 
Sep 2009
32
4
Milwaukee, Wisconsin
You are right... I read this part in one big swoop "no three of them in line such that all the distances between pairs of them are distinct" and thought it just meant no three were in a line for some reason. My bad.